Theory and Applications of Categories, Vol. 13, No. 4, 2004, pp. 61–85.
UNIVERSAL PROPERTIES OF SPAN
Dedicated to Aurelio Carboni on the occasion of his 60th birthday
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
Abstract. We give two related universal properties of the span construction. The
first involves sinister morphisms out of the base category and sinister transformations.
The second involves oplax morphisms out of the bicategory of spans having an extra
property; we call these “jointed” oplax morphisms.
Introduction
Even before they were formally introduced in [Ka], the importance of adjoint functors
was recognized in many individual cases, e.g., free groups, fraction fields, Stone-Čech
compactifications, and adjunctions on linear spaces. Any functor that has an adjoint has
many important properties; for instance, a functor with a right adjoint preserves colimits
[M2]. Of course, most functors (and even many important ones) have neither left nor right
adjoints. This motivates the introduction of profunctors [Bé2]. These are generalizations
of functors and in the bicategory Prof of categories with profunctors every functor (viewed
as a profunctor) has a right adjoint.
The reader will note that these adjunctions exist in a bicategory different from the
2-category Cat of categories in which adjunctions were originally defined. Indeed, the
usual characterization (or definition) of adjunction in terms of unit and counit satisfying
the triangle equalities is a 2-categorical concept, and adding the appropriate isomorphisms
makes it into a bicategorical one. The idea of an adjunction is fruitful enough to motivate
adding a 2-cell structure to an ordinary category, just so that one can consider adjunctions.
This has been done in several ways.
Historically, the first instance of this was probably the generalization of the concept
of “function” to that of “relation” (before categories were even invented). The idea of
relations being ordered by inclusion (i.e., as subsets of the product) also goes back long
before the invention of categories, but can be considered as providing a 2-cell structure
for the category Rel of sets and relations. In this bicategory, every function determines a
relation and as such has a right adjoint, namely the reverse relation. Furthermore, every
relation is the composite of one coming from a function with the reverse of one of these,
All three authors are suppported by NSERC grants.
Received by the editors 2004-04-21 and, in revised form, 2004-05-17.
Published on 2004-12-05.
2000 Mathematics Subject Classification: 18A40, 18D05.
Key words and phrases: Span, Π2 , Beck condition, adjoints, universal property, localizations, sinister
morphisms, jointed oplax morphisms.
c R. J. MacG. Dawson, R. Paré, and D. A. Pronk, 2004. Permission to copy for private use granted.
61
62
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
and a relation is that of a function if and only if it has a right adjoint. The notion of
relation has been extended to other categories such as groups, to generalize the element
level calculation of homology to arbitrary exact categories (cf. [Br, Hi]) and has numerous
applications [Lam, M1, P].
The two constructions which concern us here are rather more general than the above.
The first concerns the notion of span, also introduced by Bénabou [Bé1]. (The definition
and basic properties will be reviewed at the beginning of Section 1.) Bicategories of
spans have been studied in contexts such as input-feedback-output systems [KSW] and
the theory of databases [JRW]. For any category A, there is a canonical embedding
A → Span(A). It has been known for some time that Span( ) is a free construction
subject to the Beck condition. This was made precise by Hermida in [He1, Thm A.2]. In
the notation of the present paper, Hermida’s theorem states that there is an equivalence
of categories
Hom(Span(A), B) Beck(A, B).
(1)
The category Hom(Span(A), B) has homomorphisms as objects and oplax transformations as morphisms. The category Beck(A, B) has all sinister morphisms A → B that
satisfy the Beck condition as objects, and strong transformations as arrows. (Recall
that sinister morphisms were defined in [DPP] as those sending every arrow to a left
adjoint.) Moreover, under this equivalence strong transformations in Hom(Span(A), B)
correspond to sinister transformations (as in Definition 1.2) in Beck(A, B).
This result can also be viewed as a restriction of the universal property of Π2 A, the
2-category obtained by freely adding right adjoints to all arrows in A as defined in [DPP],
i.e., there is an equivalence of categories
Hom(Π2 A, B) Sin(A, B),
(2)
where Hom(Π2 A, B) is the category of 2-functors and oplax transformations, and Sin(A, B)
is the category of sinister functors, and strong transformations. Moreover, under this
equivalence strong transformations in Hom(Π2 A, B) correspond to sinister transformations in Sin(A, B).
Note that (1) above can be obtained from (2) through restriction of the notion of
morphism on the righthand side of the equation. The purpose of this paper is to describe
a suitable generalization for the notion of morphism on the lefthand side of the equation,
to obtain
JOL(Span(A), B) Sin(A, B),
(3)
where JOL(Span(A), B) consists of jointed oplax morphisms. These are normal oplax
morphisms that preserve certain composites involving adjoints (cf. Definitions 2.4 and
2.14). In future work the universal property (3) will be used to generalize the Span( )
construction to categories that may not have pullbacks, and even to bicategories.
63
UNIVERSAL PROPERTIES OF SPAN
1. Span
Spans were introduced by Bénabou [Bé1] as an example of a bicategory. Let us recall the
relevant definitions. Let A be a category with pullbacks. A span A + / B in A, i.e., an
arrow in Span(A), is a diagram A ← S → B in A. A 2-cell A
S
+
s⇓
+ S
%
9 B is a commutative
diagram
Ao
/B
S
s
Ao
/B
S
Clearly, spans from A to B with 2-cells form a category. The composition T ⊗ S of two
spans
A
S
+
/ B T+ / C
is defined using the pullback
S ×B ?T
T ⊗S =
A



S ??
??


?

??
??
B


 
T?
?
??
?
C
and ⊗ extends to 2-cells in the usual way, using the universal property of pullback. For
1
1
each A, the span A o A A A / A is the identity span IA . This defines a bicategory
Span(A).
The canonical embedding ( )∗ : A → Span(A) is a homomorphism of bicategories
where A is considered as a locally discrete bicategory.
It is defined as follows: each arrow
f : A → B of A gives a span f∗ =
Ao
1A
A
f
/B
.
The only 2-cells f∗ → g∗ are identities so ( )∗ is locally full and faithful. The span
f∗ = B o
f
A
1A
/ A is right adjoint to f∗ . Indeed, the adjunctions are
η: IA → f ∗ f∗
Ao
1A
A
δ
Ao
1A
p1 A ×B A p2
and
/A
ε: f∗ f ∗ → IB
Bo
f
A
f
/A
Bo
1B
f
B
1B
/B
/ B,
where A ×B A is the pullback of f along itself. The triangle equalities are easily verified.
Moreover, every span with a right adjoint is isomorphic to some f∗ (cf. [BCSW, CKS]).
64
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
p
q
/ B is isomorphic to q∗ ⊗ p∗ ; so Span(A) is generated by
Every span A o
S
spans coming from A and their right adjoints, but not freely. In fact, a Beck condition is
satisfied: if
h
D
k
/C
g
B
f
/A
h∗
+
C
⇒
+g∗
+
f∗
A
is a pullback, then the canonical morphism
Do
k∗ +
Bo
is an isomorphism. Indeed k∗ ⊗ h∗ can be computed as follows
1 }}
C
}~ }
D
AA 1
h }}
AA
~}}
D AA 1
AA
1 }}
D
}~ }
D AA k
AA
B
and f ∗ ⊗ g∗ as
h }}
C
}~ }
C AA g
1 ~~
AA
~~~
D AA k
AA
f
A
}}
~}}
B BB 1
BB
!
B.
To pave the way for our main theorem, we give a universal property for Span(A), which
was stated, and a proof sketched, in [He1, Thm A.2] (see also [He2, Thm 2.2]). For our
purposes a somewhat different notation will be appropriate, which we shall set out and
then use to state the universal property. We shall also give a complete proof, details of
which we shall use for our main result. First we recall (cf. [KS]):
1.1. Definition.
Let η, ε: f u: A → B and η , ε : f u : A → B be two pairs of
adjoint arrows. Two 2-cells
A
f
B
g
⇓α
h
/ A
f
/ B
AO
u
B
g
⇓β
h
/ A
O
u
/ B
65
UNIVERSAL PROPERTIES OF SPAN
are called mates under the adjunctions f u and f u if β is the composite
B
A
? ??? f
??
u 
??
 ⇓ε
?


g
/ A
⇓α
B
h
AA
>A
AA ⇓η
}}
}
AA
}
}} f AA
}} u
/ B
and (consequently) α is the composite
g
A?
A
??
?
?? ⇓η 
?
 u
f ??

B
⇓β
/ B
h
/ A
CC >
}
CC f
u }}}
CC
} ⇓ε
CC
}
!
}}
B.
Note that the notion of mates defines a bijection between 2-cells of the form α and
β as in the definition above. In this paper, we will write α∗ for β, and β∗ for α, when
appropriate.
Recall that an oplax transformation t: F → G provides, for each arrow f : A → B, a
2-cell
tA /
GA
FA
Ff
⇑tf
FB
tB
Gf
/ GB.
satisfying certain conditions described, for instance, in [Ke]. We will see that all the
transformations involved in the universal properties of Span which are not strong are
at least oplax. So we will write the strong transformations as oplax ones for which the
components are isomorphisms.
1.2. Definition.
A homomorphism of bicategories F : X → Y is called sinister if
for every arrow x: X → X in X , the image F (x): F (X) → F (X ) is a left adjoint (i.e., it
has a right adjoint F (x)∗ ). Given two sinister homomorphisms F, G : X → Y, a sinister
transformation t : F → G is a strong transformation such that for each arrow x: X → X the mates
tX /
FX
GX
O
O
(F x)∗
(Gx)∗
⇓ ((tx)−1 )∗
F X
tX / GX of the naturality isomorphisms of t,
FX
Fx
tX
/ GX
∼
= ⇓ (tx)−1
F X
tX Gx
/ GX ,
66
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
are also invertible.
The definition of sinister morphism may seem inadequate in that it might be thought
that a functoriality condition on the right adjoints and the adjunctions would be desirable.
In fact, as will be shown in Proposition 1.4, this is automatic.
1.3. Definition.
For any bicategory B let Map(B) be the bicategory of maps
in B, i.e., the objects are the same as those of B and an arrow B → B in Map(B)
is an adjunction (f , u, ε, η), with f : B → B , u: B → B, ε: f u → 1B , η: 1B → uf
satisfying the usual triangle equalities. A 2-cell (α, β): (f, u, ε, η) → (f¯, ū, ε̄, η̄) is a pair
of transformations α: f → f¯, β: ū → u which are mates of each other. Composition of
arrows is given by
(f , u , ε , η )(f, u, ε, η) = (f f, uu , ε · f εu , uη f · η).
There is an obvious 2-functor Θ: Map(B) → B which is the identity on objects and
which is locally fully faithful, so basically an inclusion.
1.4. Proposition.
as
A homomorphism F : A → B is sinister if and only if it factors
A
G
/ Map(B) Θ
/B
for some homomorphism G.
Proof. The “if” part is obvious. So assume that F is sinister. For every f : A → A ,
pick a right adjoint U (f ): F A → F A for F (f ) and adjunctions εf : F (f )U (f ) → 1F A and
ηf : 1F A → U (f )F (f ). We let G(f ) = (F f, U f, εf , ηf ). On 2-cells there is no problem
because Θ is locally fully faithful, i.e., mates are uniquely determined.
As Θ is locally fully faithful, for any A
ψf ,f : U (f f ) → U (f )U (f ) making
f
/ A
f
/ A there is a unique isomorphism
(ϕf ,f , ψf−1
,f ): (F f , U f , εf , ηf ) · (F f, U f, εf , ηf ) → (F (f f ), U (f f ), εf f , ηf f )
into an isomorphism
Gf · Gf → G(f f )
in Map(B). Similarly, there is a unique isomorphism 1G(A) → G(1A ) projecting to
ϕA : 1F A → F (1A ). Because each component of these isomorphisms uniquely determines
the other, the coherence conditions needed for G to be a homomorphism follow from those
for F .
67
UNIVERSAL PROPERTIES OF SPAN
Although this proposition presents sinister morphisms in a suitably functorial fashion,
sinister transformations t: F → F do not lift to transformations µ: G → G between
the lifted G, G : A → Map(B). This is because the components t(A) of t need not be
left adjoints. To put sinister transformations in the right perspective one needs double
categories. We will introduce the double category Map(B) in a forthcoming paper.
1.5. Remark.
In [DPP] the definition of sinister transformation was wrongly given.
What we called sinister there is no condition at all; it follows from strong naturality for
2-functors. What we called strongly sinister there is here called sinister.
Since a sinister morphism sends arrows to left adjoints, the image of a comma object
X
p2
X2
will have a mate
FX
O
F p∗2
p1
/ X1
⇓ξ
x2
F p1
x1
/ X0
/ F X1
O
F x∗1
⇓ F (ξ)∗
F X2
F x2
/ F X0 .
Explicitly, the cell (F ξ)∗ is given by the composite
F p1 ·
F p∗2
−·F (ξ)·−
ηx ·F p1 ·F p∗2
1
/ F x1 ∗ · F x1 · F p 1 · F p ∗
2
−·φ−1 ·−
∼
=
/ F x∗ · F (x1 p1 ) · F p∗
1
2
/ F x∗ · F (x2 p2 ) · F p∗ −·φ·− / F x1 ∗ · F x2 · F p2 · F p2 ∗
1
2
∼
=
−·εx2
/ F x∗ · F x2 .
1
1.6. Definition.
Let F : X → Y be a sinister morphism and assume that X has
comma objects. We say that F satisfies the Beck condition if, for every comma object
X
p2
X2
its mate
FX
O
F p∗2
F X2
p1
/ X1
⇓ξ
x2
F p1
/ X0 ,
/ F X1
O
⇓ F (ξ)∗
F x2
x1
F x∗1
/ F X0
is invertible. We call F (ξ)∗ the Beck morphism of ξ.
68
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
Let
1.7. Proposition.
p1
X
p2
/ X1
⇓ξ
X2
x1
/X
x2
0
be a comma square in X and assume that x1 has a right adjoint r; then p2 also has a right
adjoint s and the square
p1
/ X1
XO
O
s
r
X2
x2
/ X0
commutes. The Beck condition for such a comma square is automatically satisfied by any
sinister F .
Proof. Take s to be the unique arrow such that ξs = εx2
X2
x2
s
1X2
/ X0
CC
CC r
CC
CC
!
p1 !
/ X1
X
p2
! X2
⇓ξ
x1
/ X0 .
x2
We see right away that p1 s = rx2 . Also, p2 s = 1X2 ; and this will be the counit εp2 for the
adjunction.
ηp1
/ rx1 p1 rξ / r2 x2 p2 = p1 sp2 and 1p : p2 → p2 sp2 . x1
Consider the morphisms p1
2
times the first and x2 times the second commute with ξ in the sense that the outside of
the following diagram commutes
x1 p1 Kx1 ηp1/ x1 rx1 p1
x1 rξ
KKKKK
KKKKKK
εx1 p1
KKKKKK
KK ξ
x1 p 1
ξ
1
x1 p1 sp2
εx2 p2
/ x2 p 2
x2 p 2
/ x1 rx2 p2
ξsp2
N NN
N NN
NN
NN / x2 p2 sp2
Thus, by the two-dimensional universal property of comma squares, there exists a
unique η̄: 1X → sp2 such that p1 η̄ = rξ · ηp1 and p2 η̄ = 1p2 . This last equality is one of
the triangle equalities. The other, η̄s = 1s , follows from the fact that
(rξ · ηp1 )s = rξs · ηp1 s = rεx2 · ηrx2 = (rε · ηr)x2 = 1p1 s .
69
UNIVERSAL PROPERTIES OF SPAN
As homomorphisms preserve adjoints we can choose F (x1 )∗ to be F (r) and F (p2 )∗ to
be F (s) together with F of the corresponding adjunctions. Then the Beck morphism is
F p1
/ F X1
F X1
GG
w;
GG
w
GG ⇓F ηx
ww
⇓F ξ
G
1 www
F x1 GG#
F
r
w
w
/ F X0 .
F X2
F x2
F
; XFF
FF F p2
xx
F s xx
FF
x
FF
x ⇓F εp2
x
F#
x
x
F X2
which by definition of εp2 is
F X1
F
; X1GG
w;
GG ⇓F η
ww
w
w
w
x1
ww ⇓F εx1 GGGG
ww
ww
ww
ww
ww F r
F x1 G#
Fr
F X2
/ F X0
F x2
F X0 .
which is the identity. Other choices for the adjunctions would give an isomorphism.
1.8. Remark.
In this paper, we will only use Definition 1.6 for locally discrete X , in
which case comma objects are pullbacks.
1.9. Lemma.
Let F, G: A → B be homomorphisms of bicategories and t: F → G an
oplax transformation. For each adjoint pair of arrows η, ε: f u: A → B in A, the mate
(tu)∗ of tu is the inverse of tf .
Proof. Since f u, we also have F (f ) F (u) and G(f ) G(u), with adjunctions
F (f )F (u)
ϕ−1
/ F (f u) F (ε) / F (1B ) ϕB / 1F B ,
and
1F A
ϕ−1
A
F (η)
/ F (1A )
/ F (uf )
ϕ
/ F (u)F (f ) ,
and similarly for G. The composite (tu)∗ · tf is given by the following pasting diagram:
FA
FA
⇑ϕ−1
A
FA
FA
⇑F η
F (1A )
FA
Ff
/ FB
⇑ϕ
/ GB
Fu
tA
/ GA
GB
⇑Gε
GB
⇑ψB
G(1B )
G(f u)
Gu
FA
GB
⇑ψ −1
⇑tu
F (uf )
FA
tB
Gf
/ GB
GB
GB
tB
/ GB
GB
GB.
⇑tf
FA
FA
FA
FA
Ff
/ FB
70
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
By functoriality of t, this pasting is equal to
FA
FA
⇑ϕ−1
A
FA
FA
⇑F η
F (1A )
Ff
⇑ϕ
FA
/ GB
/ FB
Ff
GB
⇑t(f u)
⇑Gε
F (f u)
Fu
FA
tB
FB
⇑ϕ−1
F (uf )
FA
/ FB
⇑ψB
G(f u)
tB
/ GB
GB
G(1B )
GB
GB,
/ GB
GB
and by naturality of t, this equals
FA
FA
⇑ϕ−1
A
FA
FA
⇑F η
F (1A )
Ff
FB
⇑ϕ
FA
/ FB
Ff
tB
⇑t(1B )
⇑ψB
F (1B )
G(1B )
⇑F ε
F (f u)
Fu
FA
FB
⇑ϕ−1
F (uf )
FA
/ FB
FB
tB
/ GB
GB.
By the triangle equality, this is equal to
FA
Ff
/ FB
FB
⇑ϕ−1
B
FA
Ff
tB
/ GB
/ FB
GB
⇑t(1B )
⇑ψB
F (1B )
G(1B )
FB
/ GB
tB
GB
which equals idtB ◦F f . The fact that the other composite tf · (tu)∗ is equal to idtB ◦F f can
be proved in a similar way.
1.10. Proposition.
Let A be a category with pullbacks and B any bicategory. Then
composition with ( )∗ : A → Span(A) induces an equivalence of categories between the category Hom(Span(A), B) of homomorphisms Span(A) → B with oplax transformations,
and the category Beck(A, B) of sinister morphisms A → B satisfying the Beck condition
with strong transformations as morphisms. Under this equivalence, strong transformations
in Hom(Span(A), B) correspond to sinister transformations in Beck(A, B).
Proof. Given a homomorphism F : Span(A) → B, the composite
A
( )∗
/ Span(A)
F
/B
is sinister and satisfies the Beck condition (because ( )∗ does and F is a homomorphism).
If t: F → G is an oplax transformation, Lemma 1.9 gives that for any arrow f in A, the
mate of t(f ∗ ) is the inverse of t(f∗ ), so t ◦ ( )∗ becomes a strong transformation. Moreover,
if t: F → G is a strong transformation, Lemma 1.9 implies that the mate of t(f∗ )−1 is
t(f ∗ ) and therefore is invertible; so t ◦ ( )∗ is sinister.
This gives a functor
Hom(Span(A), B) → Beck(A, B)
71
UNIVERSAL PROPERTIES OF SPAN
which we now show to be full and faithful. Let u: F ◦ ( )∗ → G ◦ ( )∗ be a strong
transformation. We wish to extend it to an oplax t: F → G. There is no choice for t on
objects nor on morphisms of the form f∗ . But t(f ∗ ) must be the inverse of the mate of
t(f∗ )−1 and so is also uniquely determined. Finally t(f∗ ⊗ g ∗ ) is uniquely determined by
pasting t(f∗ ) with t(g ∗ ), thus
/
∗
t(f∗ ⊗ g ) =
⇑((ug)−1 )∗
∼
=
⇑uf
/
∼
=
.
/
Thus t is defined on all of Span(A). Checking that t is an oplax transformation is a
straightforward computation which we omit. If u is sinister, all the 2-cells in this calculation become isomorphisms and consequently t thus defined is a strong transformation.
Finally we must show that every sinister H: A → B satisfying the Beck condition is isomorphic to some F ◦ ( )∗ . Define F : Span(A) → B by F (A) = H(A) and
p
q
/ B ) = H(q)H(p)∗ , where H(p)∗ is a chosen right adjoint for H(p).
F( A o
S
Given a 2-cell
y S FF
p yy
yy
y
|y
A bEE
EE
E
EE
p
FFq
FF
F"
s
B
y<
yy
y
yyy q
S
,
F (s) is
H(q )εs H(p )
/ H(q )H(p ).
H(q)H(p)∗ = H(q s)H(p s)∗ ∼
= H(q )H(s)H(s)∗ H(p )∗
That F is functorial on 2-cells is another straightforward calculation involving all the
coherence conditions on H. Again, we omit the details. The Beck condition ensures that
F is a homomorphism of bicategories.
1.11. Remark. As an application of this proposition we get the fact that a cocomplete
S-indexed category can be represented by a homomorphism Span(S) → Cat (cf. [BW]).
2. Jointed Oplax Morphisms
As defined in [DPP], Π2 A is the free 2-category generated by A and right adjoints for all
f
/B)
morphisms of A. A special instance of this construction for the category 2 = ( A
was presented by Schanuel and Street in [SS]. This example shows very clearly that in
72
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
general Π2 A and Span(A) are not equivalent. The bicategory Span (2) has only three
nontrivial arrows, f∗ , f ∗ , and f∗ f ∗ as shown; and one nontrivial 2-cell ε: f∗ f ∗ ⇒ 1B .
f∗
Ai
f∗
)
B⇐
g
f∗ f ∗
(4)
However, Π2 (2) (which Schanuel and Street call Adj), has, for instance, f ∗ f∗ ∼
= 1A , and
in fact there are infinitely many non-isomorphic arrows and infinitely many 2-cells.
In this paper we are mainly interested in bicategories and only need to work up to the
level of bicategorical equivalence. So, while Π2 A, like any 2-category, is a bicategory, we
may represent it in this context by any equivalent bicategory. Many of the technicalities in
the original definition of Π2 A were only there to make horizontal composition associative
and unitary “on the nose” and make the inclusion functor ( )∗ : A → Π2 A a 2-functor.
Here we may simplify calculations considerably by choosing a bicategory from the class of
those equivalent to Π2 A which makes the relationship with Span(A) more transparent.
The bicategory we use, which we shall by abuse of notation also refer to as Π2 A, is
defined as follows:
Its objects are those of A.
An arrow A → B is a nonempty path of spans in A,
A = A0 ← S1 → A1 ← S2 → A2 · · · An−1 ← Sn → An = B
(5)
We say that this is a path of length n.
The 2-cells are represented by equivalence classes of certain commutative diagrams
called fences which look like
A A0 o
y0
A B0 o
S1
x1
T1
/ A1 o
y1
/ B1 o
S2 PPP / A2 OOOB OOO
PPP
O
OO
PPP OOOOO O OO O
x2
P
P
x3
PPPy2 OOOOO O OO O
OO
O'
P'
/ B2 o
/ B3 B
T2
T3
(6)
The yi and xj are indexed as follows. A fence from a path of length m to one of length n
is a triple (φ, yi , xj ), where
φ : {0, . . . , m} → {0, . . . , n}
is an order-preserving map such that φ(0) = 0 and φ(m) = n. For each 0 ≤ i ≤ m,
yi : Ai → Bφ(i) , with y0 = 1A and ym = 1B . As φ preserves the top element, it has a left
adjoint ψ and for each 0 ≤ j ≤ n, xj : Sψ(j) → Tj . The equivalence relation is generated
by identifying two fences if they differ only by
/ Ai o
Si+1
z
{
{
hi zzz
{
l {{{x
zz
}zz
}{{ j
/ Bj
Bj−1 o
Tj
/ Ai o
S
CC i+1
DD
CC hi
DD
CC
DD
l
CC
xj DD !
!
/ Bj
Bj−1 o
Tj
Si D
Si
and
73
UNIVERSAL PROPERTIES OF SPAN
and there exists an l factoring both parallelograms.
Horizontal composition is by concatenation and vertical composition by composing
the individual components of fences
yi , xk xψ(k) ).
(φ , yj , xk )(φ, yi , xj ) = (φ φ, yφ(i)
In a forthcoming paper we will discuss the properties of this bicategory in more detail.
For now we only need that there is a homomorphism of bicategories Υ: Π2 A → Span(A),
for which we will give a direct construction.
On objects Υ is the identity. Υ takes an arrow, represented by a path as in (5) to the
inverse limit, or generalized pullback, of the path:
S1 ×A1 S2 × · · · × Sn−1 ×An−1 Sn : A
+
/ B.
It can be computed using pullbacks. Given another path
A = B0 ← T1 → B1 ← T2 → B2 · · · Bm−1 ← Tm → Bm = B
and a fence (φ, yi , xj ) between them, Υ associates to it the morphism of spans given
by the universal property of the limit
(xj pφ∗ (j) ): ΠAi Si → ΠBj Tj
where pi : ΠAi Si → Si is the projection.
For example, for the fence (6) we get
S1 ×A1 S2
(x1 p1 ,x2 p2 ,x3 p2 )
/ T1 ×B T2 ×B T3 .
1
2
If we have a different fence (φ , hi , xj ), differing only by
/ Ai o
S
CC i+1
DD
CC hi
DD
CC
DD l
CC
xj DD !
!
/ Bj
Bj−1 o
Tj
/ Ai o
Si+1
z
{
z
{
hi zz
{{
l
zz
{{ xj
z
{
}z
}{
/ Bj
Bj−1 o
Tj
Si
Si D
with the same l factoring both parallelograms, the induced morphism will be the same
except possibly in the j th component. But the commutative diagram
Ai S i
yy
yy
y
yy
|y
y
/ Ai
Si F
FF
FF
FF
l
xj FF "
pi
Tj
o
HH
HHpi+1
HH
HH
H#
vv
vv
v
v
vv xj
vz v
Si+1
74
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
shows that in fact they are equal in the j th component as well, so that Υ is well-defined
on 2-cells of Π2 A. It is not hard to check that Υ: Π2 A → Span(A) is a homomorphism
of bicategories. Span(A) is contained in Π2 A, but not as a sub-bicategory.
Recall the definition of oplax morphism of bicategories (the vertical dual of Bénabou’s
morphism of bicategories [Bé1]). An oplax morphism Φ: A → B of bicategories takes an
object A of A to an object Φ(A) of B and for all pairs A, B of objects we have a functor
ΦA,B : A(A, B) → B(ΦA, ΦB).
For each object A we are given a morphism
ϕA : Φ(1A ) → 1ΦA
f
and for each pair of morphisms A
g
/B
/ C a morphism
ϕg,f : Φ(gf ) → Φ(g)Φ(f ).
The ϕA and ϕg,f satisfy the following naturality and coherence conditions:
• ϕA is natural in A.
• ϕg,f is natural in g and f .
• The diagram
ϕhg,f
Φ(hgf )
ϕh,gf
/ Φ(hg)Φ(f )
Φ(h)Φ(gf )
ϕh,g Φ(f )
/ Φ(h)Φ(g)Φ(f )
Φ(h)ϕg,f
commutes.
• The diagram
ϕf,1A
/ Φ(f )Φ(1A )
MMM
MMM
Φ(f )ϕA
∼
= MMM&
Φ(f 1A )M
Φ(f )1Φ(A)
commutes.
• The diagram
ϕ1B ,f
/ Φ(1B )Φ(f )
MMM
MMM
ϕB Φ(f )
∼
= MMM&
Φ(1B f )M
1Φ(B) Φ(f )
commutes.
UNIVERSAL PROPERTIES OF SPAN
75
We say that Φ is normal if all the ϕA are isomorphisms. Note that from now on all our
oplax morphisms will be assumed to be normal.
The inclusion Ψ: Span(A) → Π2 A (which forms a splitting for Υ) is an oplax morphism of bicategories. Again, on the objects it is the identity. It takes a span A ← S → B
to itself considered as a path of length one. A morphism of spans s: S → S is sent to the
equivalence class of fences determined by
Ao
/B
S
s
Ao
/B
S
(which consists only of that one fence).
2.1. Proposition.
Π2 A.
Ψ is a locally fully faithful normal oplax morphism Span(A) →
Proof.
That the morphism Ψ is locally fully faithful and normal is obvious. The
morphisms Ψ(T ⊗ S) → Ψ(T )Ψ(S) making Ψ oplax are
Ao
Ao
S
/C
S ×B T
H
ww
ww
w
ww
{w
w
p1
/Bo
HH p
HH 2
HH
HH
#
T
/ C.
The associativity and unit conditions are easily checked. This is routine except for
the non-emptiness of identity paths; we illustrate how this is handled for one of the unit
laws. We must show that
Ψ(IB ⊗ S)
Ψ(λS )
/ Ψ(S)
O
λΨ(S)
Ψ(IB )Ψ(S)
=
/ IB Ψ(S)
commutes.
When we define Span(A) we first make an arbitrary choice of pullbacks for all pairs
of morphisms but we may as well make life easier by choosing pullbacks of identities to
be identities. Then the top map of our diagram is the identity. Going around the long
way gives
/B
S @@
Ao

@@

@@

@@


/B
S ?? / B Po PPP B
PPP
??
PPP
??
1 ??
1 PPPP
P'
/B
S
1
Ao
Ao
which is the identity on S.
76
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
2.2. Proposition.
Ψ is a local left adjoint for Υ, i.e. the functors
Π2 (A)(A, B) o
ΥA,B
/
Span(A)(A, B)
ΨA,B
obtained by applying Υ and Ψ are adjoints
ΨA,B ΥA,B .
Proof. Consider an object A ← S → B of Span(A, B) and A ← T1 → B1 ← T2 →
B2 · · · Tn → B an object of Π2 (A)(A, B). In this case, a morphism ΨA,B (S) → Bj , Tj is
a diagram of the form
o
/
P
nnn A nnnnn S PPPPP B PPPPP
n
n
PPPP
n
PPP
n
nnn
PPP
nnn
···
PPPP
n
n
n
n
PPP
nn
nn
PPP
n
n
n
P
n
no
(
vn
/
o
/
/B
···
B
T
B
T
T
A
1
1
2
2
n
as there is only one choice for the indexing function and no possibility of applying the
equivalence relation. This is clearly the same as a morphism of spans
Ao
Ao
/B
S
Bj
Tj
/ B,
i.e., a morphism S → ΥA,B Bj , Tj .
The composite ΨΥ gives an oplax comonad on Π2 A which is idempotent and the identity on objects. Lax monads on bicategories were considered by Carboni and Rosebrugh
[CR] as an extension of Kock’s work on tensor products in categories of algebras over a
monoidal category [Ko]. They show that given a lax monad on a bicategory which is the
identity on objects, a new bicategory can be formed with the same objects, whose arrows
are algebras and composition given by a tensor product which classifies “bilinear” cells.
This tensor product is defined as a joint coequalizer of 2-cells which is assumed to exist
and be preserved by the monad. Our situation is dual with the 2-cells reversed. Local
equalizers don’t exist in Π2 A but as our comonad is idempotent the arrows to be equalized
are already equal (this point was already made in [RSW] in their Proposition 43 and the
remark before it), so those equalizers do exist and are preserved by the comonad. Thus
we obtain Span(A) as coalgebras for the oplax comonad ΨΥ on Π2 A. We summarize
this in the following.
2.3. Proposition.
Span(A) is equivalent to the Eilenberg-Moore category (Π2 A)ΨΥ
for the idempotent oplax comonad ΨΥ on Π2 A.
The existence of Ψ hints at another possible universal property of Span(A), namely
with respect to normal oplax morphisms. But these don’t preserve adjoints in general, so
UNIVERSAL PROPERTIES OF SPAN
77
the value of such a morphism on a span would not be determined by its values on arrows
coming from A. A closer look at Ψ shows that it preserves some compositions, in particular
those of the form f∗ ⊗ S. By this we mean that the canonical morphism Ψ(f ⊗ S) →
Ψ(f∗ )Ψ(S) is an isomorphism. We show something a bit stronger in Proposition 2.9 below.
This leads to the following definition.
2.4. Definition.
We say that a (normal) oplax morphism of bicategories Φ : A → B
is jointed if for any pair of morphisms f : A → B and g : B → C for which g has a right
adjoint, the structure cell φg,f : Φ(gf ) → Φ(g)Φ(f ) is an isomorphism.
Note that we do not require Φ to preserve composites gf where f has a left adjoint;
we will show below that this follows automatically.
Note also that jointedness can be defined identically for lax morphisms. If the structural cells ψA : Φ(1A ) → 1ΦA and (when g has a right adjoint) ψg,f : Ψ(g)Ψ(f ) → Ψ(gf )
are isomorphisms, then their inverses are isomorphisms exactly as required for a jointed
oplax morphism; the difference rests entirely in the direction of the “ordinary” structural
cells.
The usefulness of this condition has already been noted by several authors. The authors of [CKW] study morphisms of bicategories that are weaker than lax or oplax. Their
“flabby” morphisms have laxity for composition on the left, and oplaxity for composition
on the right, by maps. They observe that, when there is full oplaxity, they get what we
call jointedness. However, they only consider the locally-ordered case. In [CKVW], they
develop a very nice general theory (not just locally-ordered) in which they consider all
combinations of laxity and oplaxity. They do a detailed study of the functorial properties
of the Span construction but do not consider its universal property. In [DMS], Day, McCrudden, and Street use jointedness; Clementino, Hofmann, and Tholen have also noted
the usefulness of the property (see [CHT]).
2.5. Example.
Let R be a commutative ring and consider the contravariant functor
Hom(−, R) : R-Mod → R-Mod. There is a canonical morphism for any R-modules A
and B,
ϕA,B : Hom(A, R) ⊗ Hom(B, R) → Hom(A ⊗ B, R)
f ⊗ g −→ ( A ⊗ B
f ⊗g
/R⊗R µ
/ R ).
If we make R − Mod into a one-object bicategory M we get an oplax morphism
( )∗ : M → Mco
It is normal as R∗ ∼
= R. Moreover, it is jointed: A has a right adjoint if and only if it
is finitely generated and projective, and it is well known that in that case, ϕA,B is an
isomorphism for all B.
78
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
2.6. Example.
Closer to our discussion is the following. Let A and B be categories
with pullbacks and F : A → B an arbitrary functor. We get an oplax morphism
Span(F ): Span(A) −→ Span(B)
−→
F p F S KKF g
p S AAg
K%

zuuu
A
FA
A
F A .
The structure morphisms expressing oplaxity are given by the universal property of pullback
F (S ×ACS )
{{ ϕ
{{
{
{
{{F S × {
FA
{{
m
{{ mmmmmm
{
}{vm{mmm
p.b.
F S RRR
RRR
mmm
m
m
R
m
RRR
mmm
RR(
v mm
m
CC
CC
CC
C
FQS CCC
QQQ CC
QQQ CC
QQQ C!
(
F S RRR
m
RRR
mm
m
m
RRR
mmm
m
RR(
m
mv m
F A
F A .
Clearly, if F preserves pullbacks then all the ϕ are isomorphisms and Span(F ) is a homoFA
p
q
/ A has a right adjoint if and only if p is an isomorphism
morphism. But A o
S
and any F will preserve the pullback
S ×A SK
sss
ysss
S KKK
KK
q KK%
A
KKK
KK%
S
rr
r
r
yrrr p
when p is an isomorphism. So Span(F ) is a jointed oplax morphism.
2.7. Proposition.
Let Φ: A → B be oplax (and normal) and suppose that f u,
ε: f u → 1B , η: 1A → uf is an adjunction. If there is a 2-cell ε: Φ(f )Φ(u) → 1Φ(B) such
that
Φ(ε)
Φ(f u)
ϕf,u
Φ(f )Φ(u)
/ Φ(1B )
(7)
ϕB
/ 1Φ(B)
ε
commutes, then Φ(f ) Φ(u) with counit ε.
Proof. Let η be the unique arrow such that
Φ(1A )
Φ(η)
ϕA ∼
=
1Φ(A)
/ Φ(uf )
η
ϕu,f
/ Φ(u)Φ(f );
(8)
79
UNIVERSAL PROPERTIES OF SPAN
then
Φ(f :)
Φ(f
)
A
::
:: ∼
::=
::
::
Φ(f η)
Φ(f 1A )
ϕf,1A
/ Φ(f uf )
::
::
:: ϕf u,f
ϕf,uf ::
::
:
Nat
/ Φ(f )Φ(uf )
Φ(f )Φ(1AΦ(f
) )Φ(η)
:
Φ(f )ϕA
Φ(f )
∼
=
(8)
/ Φ(f )1ΦA
∼
= Φ(εf )
(9)
/ Φ(1B f )
ϕ1B ,f
Nat
Φ(f u)Φ(fΦ(ε)Φ(f
) / Φ(1
B )Φ(f )
)
coh
::
::
::
ϕf,u Φ(f )
Φ(f )ϕu,f ::
::
/ Φ(f )Φ(u)Φ(f )
εΦ(f )
Φ(f )η
ϕB Φ(f )
(7)
/ 1ΦB Φ(f )
∼
=
/ Φ(f )
shows one of the triangle equalities. The other triangle equality is obtained in a similar
fashion.
We say that the morphism Φ preserves the adjunction f u when the condition of
this proposition is satisfied, and Φ preserves adjoints if it preserves all adjunctions.
Jointed oplax morphisms of bicategories preserve adjoints.
2.8. Proposition.
Proof. Let the oplax morphism Φ: A → B be jointed, and let f : A → X, u: X → A,
ε: f u → 1X , η: 1A → uf be the data for an adjunction. We may take
ε = Φ(f )Φ(u)
ϕ−1
f,u
/ Φ(f u)
Φ(ε)
/ Φ(1X ) ϕX / 1Φ(X) .
If Φ preserves adjoints and is normal, then it is jointed.
2.9. Proposition.
Proof. Suppose f has a right adjoint u with adjunctions ε and η as usual. Let g be
any morphism composable with f . We will show that the composite
Φ(f )Φ(g)
Φ(f )Φ(ηg)
/ Φ(f )Φ(uf g)
Φ(f )ϕu,f g
/ Φ(f )Φ(u)Φ(f g)
εΦ(f g)
/ Φ(f g)
is inverse to ϕf,g . Indeed
Φ(f )Φ(g)
O
ϕf,g
Φ(f g)
Φ(f )Φ(ηg)
/ Φ(f )Φ(uf g)
O
Nat
Φ(f ηg)
ϕf,uf g
Φ(f )ϕu,f g
Assoc
/ Φ(f )Φ(u)Φ(f g)
O
ϕf,u Φ(f g)
/ Φ(f uf g)
εΦ(f g)
(7)
/ Φ(f g)
O
ϕB Φ(f g)
/ Φ(f u)Φ(f g)
/ Φ(1B )Φ(f g)
SSSS δf u,f g
Φ(ε)Φ(f g) kk5
SSSS Φ(εf g)
kkk
SSSS
kkkk
∆=
Nat
k
k
ϕ
SSS)
k
1
,f
B g
kkk
idΦ(f g)
/ Φ(1 f g)
B
80
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
commutes for the indicated reasons and the composite of the ϕ’s on the right is the
identity (unit law for oplax morphisms). We also have
Φ(f )Φ(g)
Φ(f )Φ(ηg)
Φ(f )ϕu,f g
/ Φ(f )Φ(uf g)
Φ(f )ϕuf,g
Φ(f )ϕ1,g
Nat
/ Φ(f )Φ(u)Φ(f g)
/ Φ(f g)
ϕf,g
Φ(f )Φ(u)ϕf,g
Assoc
εΦ(f g)
Nat
/ Φ(f )Φ(u)Φ(f )Φ(g)
/ Φ(f )Φ(g)
εΦ(f )Φ(g) 5
u,f Φ(g)
mm6
(9)
mmm
m
m
mm
mmm
mΦ(f
m
m
)ηΦ(g)
mmm
mmm
/ Φ(f )Φ(uf )Φ(g)
Φ(f )Φ(1A )Φ(g)
P Φ(f )Φ(η)Φ(g)
Φ(f )ϕ
PPP
PPP
PPP
PPP
Φ(f )ϕA Φ(g) PPPP
PPP
(
(8)
idΦ(f )Φ(g)
Φ(f )Φ(g)
and the composite of ϕ’s on the left is the identity by the unit law for oplax morphisms.
This shows the Φ preserves composites when the second arrow (diagrammatically) has a
right adjoint. Thus Φ is jointed.
We have shown that, for normal oplax morphisms, jointedness is equivalent to preservation of adjoints, which is a self-dual property (reversing arrows). This gives the following
corollary.
2.10. Corollary.
A normal oplax morphism Φ is jointed if and only if for every f
and g where f has a left adjoint, ϕg,f : Φ(gf ) → Φ(g)Φ(f ) is an isomorphism.
The following example shows that the condition that Φ be normal is necessary in
Proposition 2.9.
2.11. Example.
Let G be a comonad which is adjoint to itself, e.g.,
Ab g G G: A → A ⊕ A.
This comonad gives rise to an oplax morphism 1 → Cat which is not normal and preserves
adjoints. However, it doesn’t preserve composition with adjoints, since 1 ◦ 1 = 1, but
G◦G∼
= G.
2.12. Proposition.
Ψ: Span(A) → Π2 A is jointed.
Proof.
Let A S+ / B T+ / C with T a left adjoint, so that the h in the diagram
below is an isomorphism. Then ϕS,T is represented by the bottom fence in the following
diagram, and ϕ−1
S,T the top one.
Ao
f
Ao
g
f
k
@@


@@ h−1 g

@@



S g /Bo h T k /C
1
Ao
h
/C
S ?? / B Oo OO T
OOO −1
??
Okh
??
OOO
OOO
1 ??
O'
/C
S @ kh−1 g
f
UNIVERSAL PROPERTIES OF SPAN
81
The composite is
Ao
Ao
f
g
h
k
/C
S OOOO / B Oo OO T
OOO −1
OOO
Okh
OOO
OOO
1
OOO
−1 g OOO
h
OO'
O'
S g /Bo h T k /C
f
and h−1 factors the parallelogram, showing that this fence is equivalent to the identity.
The other composite gives the identity immediately.
The following lemma will be useful in the proof of the main theorem of this section.
2.13. Lemma.
Let G, H: X → B be jointed oplax morphisms and t: G → H an oplax
transformation. If f : X → Y is left adjoint to u in X , then the cell
GX
Gf
tX
/ HX
⇑tf
GY
tY
Hf
/ HY
is an isomorphism whose inverse is the mate of
GY
Gu
GX
tY
⇑tu
tX
/ HY
Hu
/ HX
Proof. The proof of this lemma is exactly the same as the proof of Lemma 1.9. Note
that in that proof we only use ϕ−1 and ψ −1 for composites of arrows where one of the
arrows has an adjoint, as in the definition of jointedness.
2.14. Definition.
For bicategories A and B, we will denote the category of jointed
oplax morphisms from A → B with oplax transformations by JOL (A, B).
For a category A, we let Sin(A, B) be the category of sinister morphisms from A to
B with strong transformations.
2.15. Theorem.
Let A be a category with pullbacks with canonical inclusion
( )∗ : A → Span(A),
and let B be any bicategory. Then:
1. Composing with ( )∗ gives an equivalence of categories between JOL (Span(A), B)
and Sin(A, B).
2. An oplax transformation t: G → H: Span(A) → B is strong if and only if t( )∗ is
sinister.
3. G is a homomorphism if and only if G( )∗ satisfies the Beck condition.
82
R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK
Proof. For jointed oplax G : Span(A) → B, G( )∗ : A → B is a homomorphism as
G preserves left composition by left adjoints. G also preserves adjoints and f∗ is a left
adjoint for each f in A, so G( )∗ is sinister.
For an oplax transformation t: G → H, t( )∗ is certainly an oplax transformation, but
as f∗ is a left adjoint, t( )∗ is in fact a strong transformation in view of the previous
lemma.
Note that t and t( )∗ have the same values at both the arrow and 2-cell levels, so
composing with ( )∗ does indeed give a functor
JOL(Span(A), B) → Sin(A, B).
We shall show that it is full and faithful. Let G, H : Span(A) → B be jointed oplax
morphisms and t: G( )∗ → H( )∗ a strong transformation. We must show that there is
a unique oplax transformation u: G → H such that t = u( )∗ . As the objects of A and
p
q
/ B be a morphism
Span(A) are the same, there is no problem there. Let A o
S
in Span(A). If there is such a u, then by the previous lemma, the mate of u(p∗ ) is the
inverse of u(p∗ ), or put another way, u(p∗ ) is the mate (u(p∗ )−1 )∗ of the inverse of u(p∗ ).
Thus u(S) must be given by
GA
G(p∗ )
⇑ϕ
GS
GB
tA
GA
⇑(t(p∗
GS
G(q∗ )
/ HA
/ HS
tS
⇑t(q∗ )
GB
HA
H(p∗ )
)−1 )∗
⇑ϕ−1
H(q∗ )
/ HB
tB
HS
HB.
This shows the uniqueness of u. Showing that u, thus defined, is a lax transformation is
a tedious calculation that we won’t reproduce here.
We see immediately from the formula for u(S) that u is strong if and only if t is
sinister.
Next we show that for every sinister F : A → B there exists a jointed oplax G: Span(A) →
B such that G( )∗ ∼
= F . Of course G is the same as F on objects. As jointed oplax morphisms preserve left composition by left adjoints G(q∗ ⊗ p∗ ) must be G(q∗ )G(p∗ ) and as
they preserve adjoints as well this must be F (q)F (p)∗ where F (p)∗ is a chosen right adjoint
for F (p). For a morphism of spans
Ao
Ao
p
S
r
q
/B
x
S
s
/B
G(x): G(S) → G(S ) is given by the composite
F (q)·F (p)∗
ϕ·F (p)∗
/ F (s)·F (x)·F (p)∗
F (s)·F (r)∗ ·ϕ−1 ·F (p)∗
F (s)·ηr ·F (x)·F (p)∗
/ F (s)·F (r)∗ ·F (p)·F (p)∗
/ F (s)·F (r)∗ ·F (r)·F (x)·F (p)∗
F (s)·F (r)∗ εp
/ F (s)·F (r)∗ .
UNIVERSAL PROPERTIES OF SPAN
83
Functoriality is easily verified.
To see that G is oplax consider a composite of spans,
A
S ×B ?T



S ??
p 
??q

??


p1
??p2
??



t
B
T ??
??u
??
C.
We have
F (up2 )·F (pp1 )∗
can
∼
=
/ F (u)·F (p2 )·F (p1 )∗ ·F (p)∗ Beck / F (u)·F (t)∗ ·F (q)·F (p)∗ .
That G is normal is easily seen. T has a right adjoint if and only if t is an isomorphism.
Then by Proposition 1.7, the Beck morphism is an isomorphism; so G is jointed.
It is now clear from the above formula that G is a homomorphism if and only if F
satisfies the Beck condition.
2.16. Remark.
As the above universal property of Span(A) does not involve
pullbacks or the Beck condition, we could use it to define Span for an arbitrary A or
even a bicategory A; this will be done in a forthcoming paper.
Acknowledgements. The authors thank Aurelio Carboni, Robert Rosebrugh, Dietmar Schumacher, Richard Wood (and an anonymous referee possibly disjoint from this
set) for stimulating conversations and helpful suggestions at various times during the
writing of this paper.
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Dept. of Mathematics, Dalhousie University, Halifax, NS
Email: Robert.Dawson@smu.ca, pare@mathstat.dal.ca, pronk@mathstat.dal.ca
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